Group of Rotations Stabilizer in D4

Question

In D4, the subgroup of rotations is not a stabilizer for any point in a square (even the center). Am I missing anything? Thanks!

Edit: To clarify, I wanted to ask if the subgroup of rotations of D4 is a stabilizer for any point in a square (my answer was that it is not).

Answer 1

You are right. Any non-center point is clearly not stabilized by any non-trivial rotation, thus its stabilizer cannot contain this subgroup. The stabilizer of the center is plainly seen to be the entire group (it is stabilized by both generators), and we are done.